Necessary and Sufficient Conditions for Oscillation of Neutral Equations with Deviating Arguments

Consider the first‐order neutral differential equation 1d d t[ x ( t ) + p x ( t − τ ) + r x ( t − ρ ) ] + q x ( t − σ ) + g x ( t − μ ) = 0 , where p, r, q, g , τ, p ∈ R and σ, μ ∈[0, ∞). A necessary and sufficient condition for the oscillation of all solutions of (1) is that its characteristic equation λ+λ pe −λτ +λ re −λ p + qe −λσ + ge −λμ =0 has no real roots. The method of proof has the advantage that it results in easily verifiable sufficient conditions (in terms of the coefficients and the arguments only) for the oscillation of all solutions of (1).